Find the volume of the parallelepiped & tetrahedron with one vertex at the origin and adjacent vertices are at (1, 0, -3), (1, 2,4) & (5, 1, 0).

Sure, let's start with the parallelepiped.

The volume of a parallelepiped formed by vectors a, b, and c is given by the scalar triple product of the vectors, which is the absolute value of the determinant of a 3x3 matrix formed by the vectors.

The vectors a, b, and c are given by the coordinates of the vertices:

a = (1, 0, -3) b = (1, 2, 4) c = (5, 1, 0)

The determinant of the matrix formed by these vectors is:

|a b c| = |1 1 5| |0 2 1| |-3 4 0|

= 1*(20 - 14) - 1*(00 - (-3)1) + 5(04 - 2*(-3)) = -2 - 3 + 30 = 25

So, the volume of the parallelepiped is 25 cubic units.

Now, let's find the volume of the tetrahedron.

The volume of a tetrahedron is one sixth of the volume of the parallelepiped formed by the same vectors. So, the volume of the tetrahedron is 25/6 = 4.1667 cubic units.